# Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty general and apply to Laplacian-Beltrami operator on manifolds. Their paper has bounds that are possibly too loose for my application.

Thanks, John

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I have had an analogous question but received no answer: mathoverflow.net/questions/55235/… –  András Bátkai Jan 12 at 9:02
I think that Sogge's bounds are as good as as you can hope for. In any case, the answer depends on the geometry of the boundary, and more precisely the billiard dynamics. (Think off your domain as a billiad table.) –  Liviu Nicolaescu Jan 12 at 12:34
Can you point me to the papers of Sogge that seem relevant? I see some papers involving $L_p$ bounds, but not anything on $L_\infty$. –  Ray Yang Feb 2 at 6:49
The recent work of Fanghua Lin and Kenig for div$(a\nabla u)=\lambda u$ could be of interest for you. –  Athanagor Wurlitzer Nov 25 at 9:18